A "change of basis" is an action performed in linear algebra, whereby a change in fundamental structure yields an entirely new viewpoint. This blog began as a record of a pedagogical change of basis for me, and continues as an ongoing account of my thoughts as I design and direct courses in mathematics at the University of North Carolina, Asheville.

Tuesday, May 18, 2010

From Leon Botstein's Foreward to Writing to learn mathematics and science (eds. Paul Connolly and Teresa Vilardi, 1989, New York and London: Teachers College Press, p. xiv): "The use of ordinary language in the teaching of science and mathematics enables the teacher to connect what otherwise might seem an arcane and distinct set of languages, thought processes, insights, facts, and understandings to experience, in the everyday sense that Dewey realized could constitute the basis for motivating learning, memory, and long-term comprehension."

From Paul Connolly's "Writing and the ecology of learning" (op. cit.), p. 7:

[In the all-too-typical science classroom] the important feature of education becomes saying the right words, not learning how to use one's own words. In such circumstances the "language" of science remains for many students a set of foreign words, dead as Latin, to be memorized from a book. It is not the constructive speech of a vital culture. Students then regard "definition" as a chain of words that bonds one to "truth" and "reality"; if one link is forgotten, the whole chain of understanding is broken. They have...no notion that meaning is recomposed in each new personal performance on the public instrument of language.

From Sheila Tobias's "Writing to learn in science and mathematics" (op. cit.), p. 49 (cf. my Charleston colleagues' and my recognition of the importance of visual rhetoric in the legibility of mathematical writing):

Even a cursory examination of math and science textbooks reveals that these books are not meant to be read as we understand the act of reading....As I write elsewhere, clarity in books in other subjects is achieved through repetition, using different words to restate a single idea, slowing the pace, using a spiral kind of organization that keeps coming back to the same idea at different levels, using topic and summary sentences to nail down what the paragraph contains, and always foreshadowing the point to be made later on. In math and science texts, we find, instead, pages of information with virtually no repetition, no varying of pace, few topic and concluding sentences, as few words as possible, written with the expectation that the reader will not proceed to the next sentence or point without having thoroughly mastered the one at hand.

From Thomas Kuhn's The structure of scientific revolutions (1962, Chicago: The University of Chicago Press), p.11: "Men [sic] whose research is base on shared paradigms are committed to the same rules and standards for scientific practice. That commitment and the apparent consensus it produces are prerequisites for normal science, i.e., for the genesis and continuation of a particular research tradition."

And, p. 20, on the evolution of scientific genres: "No long [after the establishment of a paradigm] will [a scientist's] researches usually be embodied in books addressed, like Franklin's Experiments...on Electricity or Darwin's Origin of Species, to anyone who might be interested in the subject matter of the field. Instead they will usually appear as brief articles addressed only to professional colleagues, the men [sic] whose knowledge of a shared paradigm can be assumed and who prove to be the only ones able to read the papers addressed to them."

Finally, p. 21: "Although it has become customary, and is surely proper, to deplore the widening gulf that separates the professional scientist from his [sic] colleagues in other fields, too little attention is paid to the essential relationship between that gulf and the mechanisms intrinsic to scientific advance."